A, B are linear transformation of vector space V such that AB=BA=E.
How do I know from AB=E that A is surjection and B is injection?

Question

A, B are linear transformation of vector space V such that AB=BA=E.
How do I know from AB=E that A is surjection and B is injection?

anonymous · Answer

I don't know if its necessarily possible, noting that either of $A,B$ are interchangeable, by the above.

gorica · Answer

Theorem says:
 If A is linear transformation of finite-dimensional vector space V(F) and $$B: V \rightarrow V$$ is mapping such that AB=E, then BA=E too, i.e. A is regular linear transformation and B is inverse linear transformation for A.

Proof says:
If AB=E, than B is injection and A is surjection. (I won't write the rest of proof, just this is what I don't get. E is identical mapping.)

anonymous · Answer

I believe that this is a "without loss of generality" case. It's impossible to know, but, we have that $E$ is a bijective identity element, hence each of $A, B$ must be at least one of injective and surjective.

anonymous · Answer

"bijective identity element" <- this statement is kind of tautological, since any identity mapping is bijective and commutative under any group, ring, or field.